Transformations of Currents in Sigma-Models with Target Space Supersymmetry

We develop a framework for systematic study of symmetry transformations of sigma-model currents in a special situation, when symmetries have a well-defined projection onto the target space. We then apply this formalism to pure spinor sigma-models, and describe the resulting geometric structures in the target space (which in our approach includes the pure spinor ghosts). We perform a detailed study of the transformation properties of currents, using the formalism of equivariant cohomology. We clarify the descent procedure for the ''universal'' deformation corresponding to changing the overall scale of the worldsheet action. We also study the contact terms in the OPE of BRST currents, and derive some relations between currents and vertex operators which perhaps have not been previously acknowledged. We also clarify the geometrical meaning of the ''minimalistic'' BV action for pure spinors in AdS.


Introduction
Consider a two-dimensional sigma-model invariant under some group of symmetries.Infinitesimal symmetries form a Lie algebra g.For each ξ ∈ g, there is a corresponding infinitesimal field transformation δ ξ .The Lagrangian is invariant up to a total derivative: (1.1) We consider the Lagrangian as a two-form on the worldsheet; α is a one-form.We use angular brackets, to emphasize that α depends on ξ linearly.Noether charge, which we denote Q⟨ξ⟩, is given by an integral of a conserved current over a space-like contour on the worldsheet Q⟨ξ⟩ = j⟨ξ⟩.
The charges transform covariantly up to constant cocycles where C⟨ξ, η⟩ is constant (i.e., field-independent).In many cases, it is true that C⟨ξ, η⟩ = 0, and we will here assume that this is the case.Then, it follows that the currents transform covariantly up to a total derivative δ ξ j⟨η⟩ = j⟨[ξ, η]⟩ + dΨ⟨ξ, η⟩. (1.2) In this paper, we will investigate various general properties of α and Ψ, and the relation between them, in one special case.This special case is when the symmetries are "projectable to the target space", in the sense which we explain now.

Symmetries projectable to the target space
The phase space is usually the cotangent bundle over the configuration space.Symmetries act in the cotangent space.In this paper, we will concentrate on a special case, when the symmetries have a well-defined projection on the configuration space.(This is not true in general, since the symmetry transformation of coordinates may depend on momenta.)Moreover, we assume that an infinitesimal symmetry ξ corresponds to a vector field v⟨ξ⟩ on the target space.(This would not be the case if derivatives of the fields were involved.)In this special case, α⟨ξ⟩ of equation (1.1) can be viewed as a one-form on the target space linearly dependent on ξ as a parameter.Similarly, the Ψ of equation (1.2) can be viewed as a function on the target space parametrized by ξ and η.We explain that α and Ψ should be thought of as defining the cochains of the bicomplex d + d Lie , where d is the de Rham differential in the target space X, and d Lie the differential of the Lie algebra cohomology complex.We derive several relations between α, Ψ, and various objects associated with them.The first observation is that there exists V , a 2-cochain of g with values in C ∞ (X), such that where W is constant on the target space; it defines a cohomology class [W ] ∈ H 3 (g, R).See Section 2.
The relation between α and Ψ is discussed in Section 5. We formulate it in terms of the "BRST model" of equivariant cohomology (see [8]): Details are described in Section 5.
As a first example, we consider the WZW model in Section 6.Then we discuss applications to the pure spinor sigma-models.

Applications to pure spinor superstring
Our study is motivated mostly by the pure spinor sigma-model of [6], and it will be our main example.We will restrict ourselves to Type II string.The corresponding sigma-model, in any background, has the following distinguishing property.There are always two anticommuting nilpotent symmetries, Q L and Q R : and two u(1) symmetries called "ghost numbers".The Q L and Q R have charges (1, 0) and (0, 1), correspondingly, under these two symmetries.

Universal vertex operator
The Lagrangian is only BRST invariant up to a total derivative This defines the "universal vertex operator" It is the unintegrated vertex operator corresponding to the overall scale of the worldsheet action.Its descent works off-shell on the worldsheet, and has a general description in any curved background.
The case of flat space is special.In this case V is BRST exact is the ghost number one vertex operator corresponding to the translation.Generally speaking, exact vertices ) correspond to infinitesimal target space diffeomorphisms.In particular, the V of equation (1.3) correspond to space-time dilatations.Although the flat space V is BRST exact, there is still an obstacle to being able to choose a BRST-invariant Lagrangian.It is a nontrivial element of -the ghost number one BRST cohomology in one-forms on the target space.While in general background the obstacle is in . See Section 10.

BRST invariance of BRST currents
While the Lagrangian is only invariant up to a total derivative, the BRST currents are BRST closed strictly (not just up to a total derivative).Moreover, they are actually BRST exact.We will give a concise proof of this in Section 8.This is a consequence of the ghost number symmetry, which acts on fields as a phase rotation.

Non-covariance of global symmetry currents
Generally speaking, currents transform covariantly only up to a total derivative.As a simple example, consider superstring in flat space in pure spinor or Green-Schwarz formulation.The supersymmetry currents are, schematically we have δ L S L ≃ ∂x and δ R S R ≃ ∂x.This is not covariant.(The covariant transformation would have P = * dx on the right-hand side.)The difference is in the total derivative dx.There is a cohomology class responsible for this non-covariance, equation (10.6).

Contact terms
The BRST currents j⟨Q L ⟩ and j⟨Q R ⟩ are holomorphic and antiholomorphic, respectively In Section 9, we study the contact term in the OPE of j L and j R and formulate a conjecture about its relation to the universal vertex operator.

Vertex operators corresponding to currents
In Sections 10 and 11, we apply our formalism to pure spinor sigma-models in flat space and AdS.Global symmetry currents in AdS are invariant under psu(2, 2|4).To each global current j a corresponds a ghost number one vertex operator Λ a .Our formalism implies some relations between them.We also clarify the geometrical meaning of the "minimalistic" B-field of [14].
In Section 12, we discuss some subtleties in taking the flat space limit of AdS.

Previous work
There have been many studies of currents in sigma-model, in particular following [1].We would especially mention [15,16,18], which have some overlap with this work.Our treatment of the worldsheet symplectic structure and global symmetries is similar to the one developed in greater generality in [9] and [7].It would be interesting to see if some results of our work could be extended to more general case (dimension higher than two and symmetries not projectable to the target space) using that more general approach.

Symmetries projectable onto the target space
Consider a classical field theory.Its phase space is T * F where F is the space of field configurations.We assume that our theory is a sigma-model.This means that field configurations are maps from the worldsheet Σ to the target space X: Suppose that the theory is invariant under a Lie group G, and the action of G comes from the action on the target space X.

2.2
The structure of Noether currents

Symplectic structure on the phase space of classical sigma-model
The symplectic potential ϑ is a one-form on the phase space and a one-form on the worldsheet, which is defined by considering the variation of the Lagrangian.Let us pick a solution of classical equations of motion, and consider its infinitesimal deformation δ as an off-shell field configuration.This means that our variation does not have to preserve the equations of motion, but the original configuration which we are varying is on-shell.Since the original configuration is on-shell, the variation of the Lagrangian is a total derivative of some 1-form on the worldsheet: Here d is the de Rham differential on the worldsheet, and δ is the de Rham differential on the phase space.Pick a space-like closed oriented contour on the worldsheet, and let ϑ mean the integral of ϑ over this contour.The "symplectic potential" (a.k.a."pδq") is defined as the restriction of ϑ to on-shell variations: The density of symplectic form is the exterior derivative of ϑ on the phase space: 2.2.2 Equivalence class of one-forms on the target space associated to a symmetry Definition of α⟨ξ⟩.Consider a symmetry ξ of the target space.If ϑ is invariant under the symmetry, then the Noether charge is the contraction of the infinitesimal symmetry vector with pδq, i.e., Q⟨ξ⟩ = pδ ξ q.In this case, the corresponding current is This formula for the current is only valid if the Lagrangian is invariant under the symmetry.However, the Lagrangian might change by a total derivative where α is some 1-form on the target space, i : Σ −→ X is the field configuration of the sigmamodel, i * α is the pullback of α to the worldsheet.The one-form α depends linearly on ξ ∈ g as a parameter.Notice that we use angular brackets α⟨ξ⟩ to emphasize that the dependence is linear.
In this case, and the current is The Noether charge is the integral of the current over a space-like contour: Indeed, In terms of the momentum variables p I , j⟨ξ⟩ = δ ξ x I p I + i * α⟨ξ⟩.
(The expansion of j⟨ξ⟩ in powers of p terminates at the linear term, because we are assuming that the symmetry is projectable to the target space.)Ambiguities in the definition of α⟨ξ⟩.The 1-form α is defined up to a total derivative and up to a symmetry variation of a one-form The ambiguity of adding d(smth⟨ξ⟩) is because the current j⟨ξ⟩ is defined only up to a total derivative.
The ambiguity of adding L ξ (smth) is because the Lagrangian L is given only up to a total derivative.Replacing L with L+dΛ with Λ ∈ Ω 1 (X) is equivalent to a canonical transformation: for any F of the form where Λ is any 1-form on the target space.This would change α⟨ξ⟩ → α⟨ξ⟩ + L ξ Λ.

Poisson bracket of charge with current
The charges satisfy But the Poisson bracket of charge with current is more complicated.Let us choose, for each ξ ∈ g, the corresponding α (i.e., "fix by hand" the ambiguity of adding d(. ..) to α), so that we have a linear map Then the charge-current Poisson bracket can be expressed through the Courant bracket: where Here the term L v⟨ξ⟩ α⟨η⟩ comes from ξ I p I , α⟨η⟩ and the term −L v⟨η⟩ α⟨ξ⟩ + dι v⟨η⟩ α⟨ξ⟩ = −ι v⟨η⟩ dα⟨ξ⟩ comes from η I p I , α⟨ξ⟩ .

Descent
The Lie algebra differential is defined as follows [10,11,12]: In particular, the deviation of the current from transforming covariantly is (2.4) Lemma 2.1.Exists V ⟨ξ ∧ η⟩ satisfying Proof .The Poisson bracket of charges is Consider field configurations which have p = 0 at τ = 0.For such configurations, equation (2.3) implies where the integral is over the spacial slice τ = 0.The image of this spacial slice in the target space can be any one-dimensional closed contour.Therefore, We observe that W is a function (0-form) and dW = 0; therefore W = const.Equation (2.6) implies that W is exact as a 3-cocycle with coefficients in C ∞ (X).But since W = const, we can as well consider W as a 3-cocycle of g with coefficients in R ⊂ C ∞ (X).Then we can ask if it is exact in the cochain complex with constant coefficients.The question is: is it possible to represent W as d Lie (. ..)where . . . is constant on X?If it is not possible, then W defines a nontrivial cohomology class with constant coefficients: [W ] ∈ H 3 (g, R).

Case [W ] = 0
In this case we can assume that d Lie V = 0, by adding a constant to V .Case H 2 (g, C ∞ (X)) = 0.In this case V = d Lie u, we can modify α → α + du, and then d Lie α = 0. Suppose that also H1 g, Ω 1 (X) = 0, then we can choose the Lagrangian to be g-invariant, and the currents g-covariant.
In Section 10.6, we consider an example where H 2 (g, C ∞ (X)) = 0 but H 1 g, Ω 1 (X) ̸ = 0.Then, the Lagrangian can not be chosen to be invariant, and the currents cannot be chosen in such a way that they would transform covariantly. 1  Case ) is an obstacle to finding an invariant Lagrangian.We may call it "generalized universal vertex operator", because vertex operators of pure spinor formalism provide an important example of this situation.The BRST transformations generate a supercommutative Lie superalgebra R 0|2 , and This is precisely the BRST cohomology at ghost number two, corresponding to physical states, therefore V is one of the physical states.This is the "universal vertex operator", see Section 8.

Ascent
Suppose that the first cohomology group of H 1 g, Ω 2 (X) = 0. Then equation (2.7) implies the existence of a two-form B such that We denote Then It is often true that H 1 g, Ω 2 (X) = 0, by some version of Shapiro's lemma [10,12].For example, this is true for the super-Poincaré symmetries in the pure spinor formalism in flat space, see Section 10.

BRST language for Lie algebra cohomology
Until this point we assumed that g is usual (not super) Lie algebra.When considering the cohomology complex of Lie superalgebras, sign rules may be confusing.Here we will suggest a formalism which automatically takes into account the ± signs.
Let M be a g-module.Let ρ denote the representation of g in M : The odd tangent space ΠT G is defined so that for all supermanifolds X: Notice that the body of ΠT G is the space of "odd loops": As a slight variation of the construction in [2], we consider the following linear space: It is possible to consider a family, parameterized by a supermanifold X: We work in a formal neighborhood of the unit of G, and M does not have to be an integrable representation.Elements of C(g, M ) are M -valued functions of the form is some odd function on X, and we use an abbreviated notation: where ζ is a Grassmann odd variable. 2It is useful to observe We may decompose into the basis of g: and introduce the "ghost" notation: Elements of C(g, M ) are in one-to-one correspondence with M -valued polynomial functions on Πg: (Here S • (. ..) denotes the space of symmetric tensors of any rank, and therefore S • (Πg) * the space of polynomial functions on Πg.)Given v, the corresponding function of the form equation (3.1) will be called Fv: The differential of the Lie algebra cohomology complex is denoted d Lie and defined as follows: These definitions work for g a Lie superalgebra.If g is a usual (not super) Lie algebra, then we can identify and our definition returns to the usual definition of Lie algebra cohomology complex.

Target space bicomplex
The central role in our considerations is played by two anticommuting differentials.One is d Liethe differential in the Serre-Hochschild complex of a Lie superalgebra.Another is d -the de Rham differential on X.Here we will discuss some basic properties of the bicomplex d + d Lie .
Consider the nilpotent operator d + d Lie acting on the cochains of g with values in differential forms on X.It is related to the de Rham operator d by a similarity transformation: where v A is the vector field on the target space corresponding to the infinitesimal symmetry.In this work, we only need differential forms which are polynomial in the differentials.We do not need pseudodifferential forms.Let us assume that the de Rham cohomology of X is R (constant functions).Then, equation (4.1) implies that the cohomology of d To prove this, let us rewrite equation (4.1) as follows: Lie .Acting by the left-hand side and the right-hand side on ω, we get Lie ω = 0. Consider the coefficient of C n+1 .Since ω is an n-form, only e C A ιv A dω contributes from e C A ιv A , d ω on the right-hand side, and equation (4.2) follows.

Transformation of currents
We will now investigate under which conditions the currents transform covariantly, i.e., (5.1)

Currents transform in an extension of the adjoint representation
Currents transform covariantly up to a total derivative: This is the definition of Ψ⟨ξ⟩⟨η⟩.This defines an extension A of the adjoint representation of g: The explicit expression for the cocycle Ψ follows from equations (2.4) and (2.5): (5.4) We will now develop a general formalism to understand equation (5.3).

Some maps between complexes
For a Lie superalgebra g, let Cg be the Lie superalgebra generated by Lie derivatives L ξ and contractions ι ξ for ξ ∈ g, see [8,Section 10] where it is called g super or g [ϵ]; it has a differential.Let M be a differential g-module, i.e., a representation of Cg with a compatible differential d M .In our case, M is the space of differential forms on X.
Consider a map ) .This map is homotopy trivial: where Indeed, using the notations of Section 3, i.e., the infinitesimal left shift of g by η.At the same time Since d Lie corresponds to ∂ ζ , equation (5.5) follows.The construction so far only requires that M is a g-module.
But once M is a differential g-module, there is yet another map which we call J:

It intertwines the two complexes:
Indeed, in notations of Section 3, It acts just on M , and does not know anything about g(ζ). 3 Notice that I+J intertwines cochain complexes with values in M and Hom(g, M ) with differential d M + d Lie .
With these notations equation (5.3) can be written as follows: In particular, if α = 0, then equation (5.1) is satisfied: If Lagrangian is invariant, then currents transform covariantly.
But the precise conditions for the covariant transformation of currents are weaker than the invariance of Lagrangian, and we will now derive it.We will start by establishing a relation to the formalism of equivariant cohomology.
3 For example,

Relation to equivariant cohomology
Various relations between V , α, W and Ψ are encoded in the complex known as "BRST model of equivariant cohomology".We will now briefly review the construction of this complex, and explain how it is useful in our context.We will start with considering another complex, known as "Weil model".This is a direct product of two complexes, and the differential is the sum of two differentials, equation (5.6).The BRST model is related to it by a similarity transformation, equation (5.7).

Weil model
Weil algebra W is a differential graded super-commutative algebra formed by generators 4 C and η with the differential We consider the space of functions of C and η with values in M , with the differential where d M is the differential of M .

BRST model
We have where d Lie is the Lie algebra cohomology differential with coefficients in Hom(S • ad g , M ).In this language I = η ∂ ∂C .The nilpotence of d Lie + d M + I + J follows from the nilpotence of d W + d M .We have where It is useful to consider the "total ghost number" defined as follows: The grading table is The term with form, C ∂ ∂C , 2η ∂ ∂η = (1, 2, 0) is missing in equation (5.8), because of equation (5.4).The nilpotence of d W + d M implies the nilpotence of d Lie + d M + I + J; this is called "BRST model" in [8]. Therefore, Since W is a constant d Lie Ψ = 0 mod const.
(By const we mean constant in the target space, i.e., independent of X, θ, λ; it only depends on C and η.) Indeed, this is necessary for the consistency with equation (5.2).Equation (5.8) is not completely general.Without changing the range of gradings on the right-hand side, we can generalize it as follows: where φ ∈ C 0 g, Hom ad g , M 0 = Hom ad g , M 0 .
The gradings on the right-hand side are still (0, 3, 0), (0, 1, 2) and (1, 0, 2).But φ can be removed by adding an exact expression.Indeed, given ψ ∈ C 1 g, M 0 , the exact expression is: This corresponds to the ambiguity of adding a total derivative to α.

Obstacles to covariance of currents
Equation (5.9) implies that the first obstacle to the covariance of currents is [IW ] ∈ H 2 (g, Hom(ad g , R)). (5.10) If [IW ] = 0, then we correct Ψ by adding to it a constant, so that d Lie Ψ = 0.In that case, the next obstacle is [Ψ] ∈ H 1 (g, Hom(ad g , C ∞ (X))). (5.11) In many situations, this cohomology group is zero due to Shapiro's lemma, and currents transform covariantly.
To summarize, the condition of the covariance of the currents is that we can choose Ψ = const.When this is so, equation (5.3) implies (5.12) This follows from equation (2.3) and equation (5.1) symmetrized under ξ ↔ η.When equation (5.12) holds, equation (5.1) becomes equivalent to This means that we can choose V as follows: To summarize, here are the conditions for the currents transforming covariantly, i.e., δ ξ j⟨η⟩ = j⟨[ξ, η]⟩: where C⟨ξ • η⟩ is constant, (5.13) where ξ • η stands for symmetric tensor product, and V is from equation (2.5).

Asymmetric covariance condition
It is also possible to formulate a weaker condition.We can ask that for some two symmetries ξ and η: This weaker condition follows directly from equations (2.3) and (2.5): V ⟨ξ ∧ η⟩ = −ι v⟨η⟩ α⟨ξ⟩ mod const.

Case of WZW model
Here we will relate our considerations to the approach of [17] to WZW model.The target space is the group manifold, and the symmetry is some combination of left and right shifts.The non-invariance of the Lagrangian is due to the B-field not being strictly invariant: A special role in [17] is played by the one-forms λ a such that Remember that α⟨ξ⟩ is defined up to d(. ..).Equations (6.1) and (6.2) imply that we can choose α as follows: Therefore, The obstacle to gauging in [17] was the right-hand side of equation (6.4), and is therefore very similar to equation (5.13).All currents in WZW model do transform covariantly, therefore both equations (5.13) and (5.14) are satisfied.However, only a subgroup can be gauged, where the constant on the right-hand side of equation (5.13) is zero.The obstacle to gauging is C⟨ξ • η⟩.
It was shown in [17] that

Now we have
The last two terms actually cancel: and therefore In the language of ghosts, Then, equation (4.2) implies that We have where λ is from equation (6.3), considered as element of C 0 g, Hom g, Ω 1 (X) , and other elements are We have IΨ + Jλ = 0. (6.8) Notice that H + λ is equivariantly closed (i.e., annihilated by d + J) iff Jλ = 0. Equation (6.8) implies that this is equivalent to Under this condition, also W + Ψ is d Lie + I-closed.Equation (6.7) implies that this is equivalent to the vanishing of the obstacle to gauging in [17]: Equation (6.5) is the equivariant analogue of the WZW transgression relation d with d + J and d Lie with d Lie + I. Notice that equation (6.5) does not require the cancellation of anomaly in the sense of [17].In fact, Jλ cancels with IΨ, equation (6.8).
7 Pure spinor sigma-model In this section, we will give a very brief review of the pure spinor worldsheet sigma-model Most of the applications in this paper are to the Type II pure spinor superstring sigmamodel [3,6].This sigma-model can be constructed for any consistent Type II supergravity background.The set of fields includes some matter fields taking values in a supermanifold (space-time), and the so-called "pure spinor ghosts".The pure spinor ghosts are called λ α L and λ α R , they are constrained to take values in a cone: where Γ m αβ are the ten-dimensional gamma-matrices.In special backgrounds, such as flat space and AdS 5 × S 5 , the sigma-model has global symmetries, basically the isometries of the background.But, crucially, there are two fermionic symmetries present at any background.They are denoted Q L and Q R and called "BRST symmetries".They are both nilpotent and anticommuting: Moreover, the Noether current corresponding to Q L can be chosen to be holomorphic, and the Noether current corresponding to Q R antiholomorphic.They are In flat space,

Universal vertex operator
Pure spinor superstring in any curved background has a natural deformation, which is the change of the overall scale of the Lagrangian.It can be thought of as varying the string tension 1 α ′ .In this section, we will explain how this "universal vertex operator" is related to α⟨Q⟩, equation (8.6).

Definition for arbitrary curved background
Equation (2.5) implies An important role is played by the ghost number symmetry.In other words, we assume that X is a graded supermanifold of degree one.(Where x and θ are of degree zero, and λ of degree one.)Let N be the grading vector field and f 0 be a function of degree zero (i.e., a function of x and θ only).A direct examination of the sigma-model Lagrangian in [6] shows that it is exactly invariant under N (since it is just a phase rotation of ghosts and their momenta).Therefore, Using the freedom of adding a total derivative to α⟨Q⟩, we can choose α⟨Q⟩ so that Also notice that we can choose V (N, Q) = 0.This is because N is just a phase rotation, therefore Therefore, In other words, both j⟨Q L ⟩ and j⟨Q R ⟩ are BRST exact.Taking into account that L N V = 2V , equation (8.1) implies Given equation (5.3), this is consistent with BRST currents being themselves BRST invariant, in any background.As we know, they are both BRST exact, equations (8.3) and (8.4).
To summarize: We must stress that this descent procedure does not require the use of the worldsheet equations of motion.The second step, equation (8.6), involves purely geometrical objects in the target space: vector field Q and one-form α⟨Q⟩.This works for all pure spinor backgrounds.

Generalization
Suppose that g is a graded Lie superalgebra.The grading operator will be denoted n : g → g.
Let us assume that n is diagonalizable, and all its eigenvalues nonzero in particular, exists n −1 .Moreover, we will assume that there is a vector field N on X, a symmetry of the Lagrangian, such that We can modify α as follows: Then the modified α satisfies We have Suppose that equation (5.13) is satisfied, then equation (8.7) implies This is a generalization of equation (8.5).If we assume that all eigenvalues of n are positive, then it follows that V is a cocycle: If its cohomology class in H 2 (g, C ∞ (X)) is nonzero, then it is a generalization of the universal vertex operator.

Contact terms
We will here study the contact terms in the OPE of left and right BRST currents.These contact terms do not have classical analogue.

BRST currents and universal vertex
Since j⟨Q L ⟩ is holomorphic and j⟨Q R ⟩ is antiholomorphic, their OPE is purely a contact term: Since both currents are exactly BRST invariant, it follows: We conjecture that V is in fact the universal vertex operator.(In particular, this implies that the universal vertex operator has ghost number (1, 1).) In flat space-time, the contact term is somewhat ill-defined, since the conserved currents are only well-defined up to terms which vanish on-shell.However, in the general curved space-time, there is no ambiguity.Indeed, all equations of motion have the conformal dimension (1, 1), except those for pure spinor ghosts, which have dimension (1, 0) or (0, 1) (i.e., The only possibility would be adding something proportional to ∂ + λ R to j⟨Q L ⟩, and something proportional to ∂ − λ L to j⟨Q R ⟩.But that would have wrong ghost number. We computed the contact terms in flat space (with the most economical definition of the currents) and in AdS 5 × S 5 .In both cases, the coefficient of δ 2 (z, z) in the contact term is the universal vertex operator.The easiest way to compute them is to use On the other hand, this is equal to In the second term, Q R j + ⟨Q L ⟩(z, z) is zero on-shell.Whenever something is zero on-shell, there is a corresponding vector field on the space of fields.In particular, the vector field corresponding to

Case of flat space
The contact term is due to the contraction of ∂ + x and ∂ − x in equations (7.1) and (7.2): 10 Pure spinor sigma-model in flat space 10.1 Symmetries of the pure spinor sigma-model in flat space The (manifest) symmetries of the pure spinor sigma-model in flat space are: The super-Poincaré group SP which is the semi-direct sum of the supersymmetry group (translations and supersymmetries) and the Lorentz group.The Lie superalgebra of the super-Poincaré group will be denoted sP (notation: sP = super-Poincaré) Two odd nilpotent symmetries Q L and Q R .The corresponding ghosts will be called χ L and χ R .
The Lie superalgebra cohomology differential is Explicitly, each part is

Supersymmetries
The infinitesimal supersymmetry transformations are The corresponding one-form of equation (2.1) is

BRST transformations
The BRST transformation is the following vector field on the space of worldsheet fields: The corresponding one-form of equation (2.1) is It can be represented as where Therefore, ι 2 Q db = 0 and equation (5.15) implies that equation (5.14) is satisfied and therefore the BRST current is BRST closed: This is also true for general curved backgrounds, and is a consequence of the ghost number symmetry, see equation (8.5).

Ascending in form rank
Because of Shapiro's lemma, L ξ H = 0.

Descending in form rank
We will use the Faddeev-Popov language.Let γ α L , γ α R be the ghosts for left and right supersymmetries, c µ and c [µν] the ghosts for translations and rotations.We will also consider the BRST symmetries.The ghosts for BRST symmetries will be denoted χ L and χ R .The descent goes as follows: The last line of V contains (as the coefficient of χ L χ R ) the universal vertex operator of flat space, i.e., (λ BRST , it does not involve χ L and χ R .In fact, it could not possibly contain χ L and χ R , because it is constant (does not depend on X, θ L , θ R ).The degree of χ equals the ghost number.Expressions with nonzero ghost number can not be constant.

Relation between H = db and W
Now we have the relation and its equivariant analogue Such relation exists because b exists because of equation (10.3).It is limited to sP ⊂ sP ⊕ R 0|2 , because there is "no b for Q".There is no such b which would satisfy 10.5 Non-covariance of sP currents Let us use The first follows from the BRST closedness of the BRST current, and the second from IW ⟨Q • Q⟩ = 0. Also, we use We hope that the notations are not too confusing; C sP is the ghost for sP, and | sP means that the argument (the η) is restricted to sP ⊂ sP ⊕ R 0|2 .It implies

This implies that Ψ⟨χ
In other words, it is an intertwiner between the adjoint of sP and the ghost number one BRST cohomology.This intertwiner associates to each element of sP the corresponding ghost number one vertex operator.It is enough to compute Ψ⟨Q⟩ on an infinitesimal translation, i.e., η = ∂ ∂X m .In this case, the IV in Ψ = IV − Jα does not contribute, and we are left with −Jα which gives Instead of being BRST closed, they satisfy 10.6 Cohomology of Q in one-forms The last term in equation (10.5) is BRST exact: This implies that we can add a total derivative to α⟨Q⟩ so that the modified (improved) α⟨Q⟩ is BRST-closed.It is given by the following expression, which contains "bare" X m : We still can not make the Lagrangian BRST-invariant, since α impr represents a nontrivial cohomology class: Moreover, the currents do not transform covariantly.Indeed, the right-hand side of equation (2.4) becomes dι Q α impr ⟨Q⟩ with In a general curved background, the universal vertex operator is a nontrivial element of H 2 (Q, C ∞ (X)).But flat space is a special case.What would be a ghost-number-two universal vertex is now BRST exact.Instead, what prevents the Lagrangian from being BRST invariant is a nontrivial element of H 1 Q, Ω 1 (X) .

Covariance with respect to a smaller subalgebra
We can achieve covariance with respect to a smaller subalgebra h ⊂ sP, if the restriction of W to h vanishes.For example, let h be the subalgebra preserving the half-BPS state, then W vanishes.In fact, This implies Then, the modified Lagrangian L − f is h-invariant.
We will now show this explicitly.We will denote X ± = X 0 ± X 9 .We will choose the momentum of the half-BPS state in direction X + , then the following bosonic generators annihilate the state: ∂ ∂X − , ∂ ∂X i , where i runs from 1 to 8. The ten-dimensional gamma-matrices are We have to restrict The first line of equation (10.5) becomes It is d Lie -exact: where  Therefore, The currents of the global symmetries are only Q-invariant up to a total derivative: Qj a = dΛ a , Λ a = d STr t a g −1 (λ 3 − λ 1 )g .
Here QL is the pullback of an exact two-form from the target space, and ι ξ QL is (by a slight abuse of notations) the pullback of its contraction with ξ.Therefore, ι ξ QL = Qj⟨ξ⟩.

"Minimalistic" B-field
According our general scheme, d(χ L α⟨Q L ⟩ + χ R α⟨Q R ⟩) is the variation of the Lagrangian under We would like to stress that the Lagrangian can not be seen as a geometrical object on the target space.The Lagrangian is not a pullback of a differential form on the worldsheet, as it contains terms like dX ∧ * dX and p ∂X. But, one can ask: is it possible to find such a 2-form B on the target space that its Lie derivative along χ L Q L + χ R Q R be equal to that total derivative?The answer is negative, in fact d(χ L α⟨Q L ⟩ + χ L α⟨Q L ⟩) is a nontrivial element of H 2 R 0|2 BRST , Ω 2 (X) .But, it is possible to find such B if we do not require it to be smooth.In fact, Here P 31 is some projector, which is a rational function of λ L and λ R .See [14] for details.We will here list some properties of B and H = dB: where This can be extended to the full g = R 0|2 ⊕ psu(2, 2|4), giving an analogue of equation (6.5):
10.5.1 sP currents are not sP-covariant This is because IW is not zero in H 2 (g, Hom(ad g , R)), see equation(5.10).This is different from the case of WZW model.In the case of W ZW model, IW is zero in H 2 (g, Hom(ad g , R)), due to the existence of the invariant C, see equation (6.7).10.5.2 sP currents are not BRST closedSince W is only nontrivial in H 3 (sP) ⊂ H 3 sP ⊕ R 0|2BRST , the corresponding component of IW is zero.In this case, the next obstacle, equation(5.11), is nontrivial: